Question

according to the centers of disease control, about 20% of american young people between the ages of 12 and 19 are obese.
a student claims that the students in his local schools are in better shape than the national average, so their proportion of obese students is less than 20%.
which are the appropriate null and alternative hypotheses for this study?
$\mathrm{h}_0: p = 20\\%, \mathrm{h}_a: p > 20\\%$
$\mathrm{h}_0: p \
eq 0, \mathrm{h}_a: p = 0$
$\mathrm{h}_0: p = 20\\%, \mathrm{h}_a: p \
eq 20\\%$
$\mathrm{h}_0: p = 20\\%, \mathrm{h}_a: p < 20\\%$
suppose that the result is to fail to reject $\mathrm{h}_0$.
which is the proper conclusion?
there is not sufficient evidence to support the claim that the students at his local schools have a smaller proportion of obese students.
there is sufficient evidence to support the claim that the students at his local schools have a smaller proportion of obese students.
there is not enough information to answer this question.

Explanation:

Part 1: Hypotheses

Step1: Define Null Hypothesis

The null hypothesis (\(H_0\)) is a statement of no effect or no difference, assuming the national proportion (20%) holds. So \(H_0: p = 20\%\).

Step2: Define Alternative Hypothesis

The student claims the local proportion is less than 20%, so the alternative hypothesis (\(H_A\)) is \(p < 20\%\).

Failing to reject \(H_0\) means we don't have enough evidence to support the alternative hypothesis (that local proportion is less than 20%). So there's not sufficient evidence for the student's claim.

Answer:

\(H_0: p = 20\%, H_A: p < 20\%\) (the fourth option: \(H_0: p = 20\%, H_A: p < 20\%\))

Part 2: Conclusion